Ironically, had I not wound up in tech as a day job, I would never have discovered Knuth's and especially Wilf's masterful books on generating function theory, which make clear just how Euler's generation worked their magic (formal manipulation of generating functions).
What even the ultrafinitists seem to fail to articulate is that the mathematical notion of infinity is just a primitive expression of an iterated calculation, i.e. of an approximative calculation that can be iterated indefinitely. And the notion of infinitesimal, in actual practical usage, is merely a placeholder for a small approximation which can be taken as small as the data allows--even to this day, nobody can beat dx and dy for setting up a differential equation.
The notion of infinity as representing a non-terminating iterative calculation clarifies the absurdity of Cantor's diagonalization argument, which turns on completing one infinite task (enumerating all numbers as binary decimals) before beginning a second infinite task (flipping one digit in the nth term of each number in the "completed" list). At each finite step, given a list of n! possible numbers of length n, this algorithm clearly fails.
Ah, it's already past 9, time for work...
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